Club president Josu Urrutia claims a firm bid for the 24-year-old was lodged on Thursday and immediately dismissed.
And Urrutia says the only way Herrera will leave the club is if he demands a move and his release clause - reported to be £30.75m - is met.
Speaking at a press conference, reported on Athletic's Twitter feed on Friday, he said: "We have had an offer for Ander Herrera. Our club is different, it is based on sentiment. Our objective isn't to make money.
"We received the offer last night and have communicated that we will not negotiate for our players.
"For a player to go, first the player must express his desire to leave the club and pay the release clause."
David Moyes, asked about Herrera at his press conference ahead of Sunday's game against Liverpool, was not prepared to name United's targets.
He said: "We would try not to disclose our bids, if that's possible. We would just let the clubs deal with it and try to do it in the professional manner."
On the difficulty of signing players in general, Moyes added: "It was going to be a struggle because I was coming into a new job and was getting to know the players.
"There was a new chief executive at the club as well so it was always going to take it a little bit of time to settle down. But we have been in and around the players we wanted to sign.
"We have been after two or three players and tried to go for them. Some of them we are still in with a shout with and some we are not. It might be busy in the next few days and I hope we do a little bit of business but I couldn't guarantee it."